Diffuse texture baking wrong
When I bake this from the provided blend file, it works.  As such this doesn't directly answer the question, but it's too large for a comment and might be helpful so I will post it anyway - if a proper &quot;here's what you did wrong and here's how to fix it&quot; answer shows up and this no longer seems useful please feel free to delete it.
Here are the steps I took from loading the blend file to having a baked output png saved and the chameleon displaying properly in cycles mode using the texture rather than the nodes:

In the node view of the 'Procedural' material, add a texture setup to receive the bake.  This texture is blank, and in this case is a 1024x1024 black png file which I saved to disk so it existed (blender is sometimes a bit weird about having its files exist before writing to them, also, cycles seems to render from-file rather than from-memory).  It's fine that this isn't connected to anything at this stage 

Select the texture node.  Just click on it so the BakedOutput node has a white outline.  This step is important because otherwise it might put the bake somewhere else - probably on top of the chameleon skin texture or something equally unhelpful.

Change the render engine to Cycles.

Set up for diffuse color baking with no lighting information.  I used these settings: 

Press Bake, and wait for it to complete

Save the resulting BakedOutput.png or whatever else you have called it.  First because it's good practice and it's easy to lose the bake if you don't, but also, perhaps more importantly, because as mentioned Cycles likes to render with the black blank texture if that's what's on disk.

Attach the texture node to the Color input in the Procedural material to check it works - you may wish to use the texture elsewhere instead, and preserve the Procedural material and just switch to the new one, but I didn't do that, I just swapped the Color input to be faster and more sure it would work.  Screenshot shows disconnected procedural result and connected texture node: 


As to what you did that caused your issue I am unsure.  I cannot seem to reproduce it - even baking with lighting information included just produces a badly lit output, rather than the mis-arranged image with the wrong parts mapped shown in the question.  I attempted baking with incorrect texture coordinates but that didn't seem to do it either.  There's only one UVMap provided so I couldn't bake using the wrong one of those... I'm not sure if I can answer the question, so feel free to ignore this answer if it isn't helpful.  Perhaps try recording your own steps and then following mine and see if we get the same result.
I -did- notice that the bake is different (and worse looking, much further away from original procedural output) baked from Edit mode instead of Object mode.  I suspect this might have something to do with it, but it still doesn't match your original image.  Perhaps the pose or armature is involved.